Recursion and Dynamic Programming

Coin Change (Bottom-Up)

Build a one-dimensional table where each amount stores the fewest coins needed to make it.

Algorithm

@steps

  1. Initialize dp[0] = 0 and all other amounts to an unreachable sentinel.
  2. Scan amounts from 1 through 6.
  3. For each coin, read the earlier cell dp[amount - coin] when it exists.
  4. Write the smallest candidate into the current amount.
  5. Print both the final answer and the full DP array. @end @complexity
  • Time: O(target * coin_count)
  • Space: O(target) @end
bottom-up dynamic programming `dp[a]` is solved from already-computed smaller amounts, so every table cell has a visible dependency.

C# DSA Implementation

basic.cs 
using System;
using System.Linq;

int[] coins = {1, 3, 4};
int target = 6;
int inf = target + 1;
int[] dp = Enumerable.Repeat(inf, target + 1).ToArray();
dp[0] = 0;
for (int amount = 1; amount <= target; amount++) {
  foreach (int coin in coins) {
    if (amount >= coin) {
      int candidate = dp[amount - coin] + 1;
      if (candidate < dp[amount]) dp[amount] = candidate;
    }
  }
}
Console.WriteLine(dp[target]);
Console.WriteLine("[" + string.Join(", ", dp) + "]");

@end @output 2 [0, 1, 2, 1, 1, 2, 2] @end